Unbiased Estimation and Variance Bounds - Estimation - Statistics Previous Year Questions

1

2019 · Statistics · Estimation · Unbiased Estimation and Variance Bounds

Statistics (ST) 2019

Let \(X\) be a normal random variable having mean \(\theta\) and variance 1, where \(1 \leq \theta \leq 10\). Then \(X\) is

A

sufficient but not complete.

B

the maximum likelihood estimator of \(\theta\).

C

the uniformly minimum variance unbiased estimator of \(\theta\).

D

complete and ancillary.

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2

2019 · Statistics · Estimation · Unbiased Estimation and Variance Bounds

Statistics (ST) 2019

Consider the model \(Y_i = \beta + \epsilon_i\), where \(\epsilon_i\)'s are independent normal random variables with zero mean and known variance \(\sigma_i^2 > 0\), for \(i = 1, ..., n\). Then the best linear unbiased estimator of the unknown parameter \(\beta\) is

A

\(\frac{\sum_{i=1}^n (Y_i/\sigma_i^2)}{\sum_{i=1}^n (1/\sigma_i^2)}\)

B

\(\frac{\sum_{i=1}^n Y_i}{n}\)

C

\(\frac{\sum_{i=1}^n (Y_i/\sigma_i)}{n}\)

D

\(\frac{\sum_{i=1}^n (Y_i/\sigma_i)}{\sum_{i=1}^n (1/\sigma_i)}\)

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3

2020 · Statistics · Estimation · Unbiased Estimation and Variance Bounds

Statistics (ST) 2020

Let X1, ..., Xn be a random sample of size n (≥ 2) from N(θ, 2θ2) distribution, where θ ∈ (0, ∞). Which of the following statements is TRUE?

A

1/(n+2) ∑(Xᵢ−1)² is the unique unbiased estimator of θ² that is a function of minimal sufficient statistic

B

(3/(n+1)) ∑Xᵢ² is an unbiased estimator of θ²

C

There exist infinite number of unbiased estimators of θ² which are functions of minimal sufficient statistic

D

There does NOT exist any unbiased estimator of θ(θ+1) that is a function of minimal sufficient statistic

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4

2022 · Statistics · Estimation · Unbiased Estimation and Variance Bounds

Statistics (ST) 2022

Let \(S^2\) be the variance of a random sample of size \(n > 1\) from a normal population with an unknown mean \(\mu\) and an unknown finite variance \(\sigma^2 > 0\). Consider the following statements:
(I) \(S^2\) is an unbiased estimator of \(\sigma^2\), and \(S\) is an unbiased estimator of \(\sigma\).
(II) \(\left(\frac{n-1}{n}\right)S^2\) is a maximum likelihood estimator of \(\sigma^2\), and \(\sqrt{\frac{n-1}{n}} S\) is a maximum likelihood estimator of \(\sigma\).
Which of the above statements is/are true?

A

(I) only

B

(II) only

C

Both (I) and (II)

D

Neither (I) nor (II)

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5

2022 · Statistics · Estimation · Unbiased Estimation and Variance Bounds

Statistics (ST) 2022

Let \(0, 1, 1, 2, 0\) be five observations of a random variable \(X\) which follows a Poisson distribution with the parameter \(\theta > 0\). Let the minimum variance unbiased estimate of \(P(X \le 1)\), based on this data, be \(\alpha\). Then \(5^4\alpha\) (in integer) is equal to ______

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6

2022 · Statistics · Estimation · Unbiased Estimation and Variance Bounds

Statistics (ST) 2022

Consider a gamma distribution with the probability density function \[ f(x; \beta) = \begin{cases} \frac{1}{24 \beta^5} x^4 e^{-x/\beta}, & x > 0, \\ 0, & \text{elsewhere}, \end{cases} \] with \(\beta > 0\). Then, for \(\beta = 2\), the value of the Cramer-Rao lower bound (rounded off to one decimal place) for the variance of any unbiased estimator of \(\beta^2\), based on a random sample of size 8 from this distribution, is ________

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7

2023 · Statistics · Estimation · Unbiased Estimation and Variance Bounds

Statistics (ST) 2023

Let \(X_1, X_2, \ldots, X_n\) be a random sample of size \(n\) from a population having uniform distribution over the interval \((\frac{1}{3}, \theta)\), where \(\theta > \frac{1}{3}\) is an unknown parameter. If \(Y = \max\{X_1, X_2, \ldots, X_n\}\), then which one of the following statements is true?

A

\((\frac{n+1}{n})(Y - \frac{1}{3}) + \frac{1}{3}\) is an unbiased estimator of \(\theta\)

B

\((\frac{n}{n+1})(Y - \frac{1}{3}) + \frac{1}{3}\) is an unbiased estimator of \(\theta\)

C

\((\frac{n+1}{n})(Y + \frac{1}{3}) - \frac{1}{3}\) is an unbiased estimator of \(\theta\)

D

\(Y\) is an unbiased estimator of \(\theta\)

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8

2023 · Statistics · Estimation · Unbiased Estimation and Variance Bounds

Statistics (ST) 2023

Let \( X_1, X_2, \ldots, X_{10} \) be a random sample of size 10 from a population having \( N(0, \theta^2) \) distribution, where \( \theta > 0 \) is an unknown parameter. Let \( T = \frac{1}{10} \sum_{i=1}^{10} X_i^2 \). If the mean square error of \( cT \) (c > 0), as an estimator of \( \theta^2 \), is minimized at \( c = c_0 \), then the value of \( c_0 \) equals

A

\(\frac{5}{6} \)

B

\(\frac{2}{3} \)

C

\(\frac{3}{5} \)

D

\(\frac{1}{2} \)

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9

2023 · Statistics · Estimation · Unbiased Estimation and Variance Bounds

Statistics (ST) 2023

Let \(\left\{-1, -\frac{1}{2}, 1, \frac{5}{2}, 3\right\}\) be a realization of a random sample of size \(5\) from a population having \(N\left(\frac{1}{2}, \sigma^2\right)\) distribution, where \(\sigma > 0\) is an unknown parameter. Let \(T\) be an unbiased estimator of \(\sigma^2\) whose variance attains the Cramer-Rao lower bound. Then based on the above data, the realized value of \(T\) (rounded off to two decimal places) equals ______________

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10

2024 · Statistics · Estimation · Unbiased Estimation and Variance Bounds

Statistics (ST) 2024

Let X_1, X_2, ..., X_n be a random sample of size n (≥ 2) from a population having Poisson distribution with mean λ, where λ > 0 is an unknown parameter. If T_1 = X̄ and T_2 = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (X_i - X̄)^2}, where X̄ = \frac{1}{n} \sum_{i=1}^n X_i, then which of the following statements is/are true?

A

T_1 is an unbiased estimator of λ

B

T_2 is an unbiased estimator of √λ

C

T_2^2 is an unbiased estimator of λ

D

Both T_1 and T_2 are estimators of λ as well as λ^2

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11

2024 · Statistics · Estimation · Unbiased Estimation and Variance Bounds

Statistics (ST) 2024

Let X_1, X_2, X_3 be a random sample of size 3 from a population having Bernoulli distribution with parameter p, where p ∈ (0,1) is unknown. Define T_1(X_i, X_j, X_k) = X_i - X_j(1 - X_k), T_2(X_i, X_j, X_k) = \frac{1}{2}(X_i + X_j), for i,j,k = 1,2,3; i ≠ j ≠ k. Let x_1, x_2, x_3 denote realizations from the random sample. Then which of the following statements is/are true?

A

T_1(X_1, X_2, X_3) has the same distribution as T_1(X_2, X_3, X_1), but T_1(x_1, x_2, x_3) ≠ T_1(x_2, x_3, x_1) for some realization x_1, x_2, x_3

B

T_2(X_1, X_2, X_3) and T_2(X_3, X_2, X_1) are both unbiased estimators for p^2

C

T_1(X_1, X_2, X_3) and T_1(X_2, X_3, X_1) are both unbiased estimators for p^2 and T_1(x_1, x_2, x_3) = T_1(x_2, x_3, x_1) for all realizations x_1, x_2, x_3

D

T_2(x_1, x_2, x_3) = T_2(x_3, x_2, x_1) for all realizations x_1, x_2, x_3

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12

2025 · Statistics · Estimation · Unbiased Estimation and Variance Bounds

Statistics (ST) 2025

Let \( X_1, ..., X_5 \) be a random sample from \( N(\theta, 6) \), where \( \theta \in \mathbb{R} \), and let \( c(\theta) \) be the Cramer-Rao lower bound for the variances of unbiased estimators of \( \theta \) based on the above sample. Then \( 15 \inf_{\theta \in \mathbb{R}} c(\theta) \) equals ______________ (answer in integer).

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13

2026 · Statistics · Estimation · Unbiased Estimation and Variance Bounds

Statistics (ST) 2026

Let \(X_1\) and \(X_2\) be independent and identically distributed random variables following normal distribution with mean \(\theta \in (-\infty, \infty)\) and variance 1. Then which of the following estimators of their expected values attains the Cramer-Rao lower bound?

A

\(X_1 + X_2\)

B

\(\frac{X_1^2 + X_2^2 + 5}{2}\)

C

\((X_1 + X_2)^2\)

D

\(2X_1 - X_2\)

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14

2026 · Statistics · Estimation · Unbiased Estimation and Variance Bounds

Statistics (ST) 2026

Let \(X_1, X_2\) be a random sample from a distribution having the population density function \[ f(x) = \begin{cases} \frac{1}{\theta} & \text{if } 0 < x < \theta \\ 0 & \text{otherwise}, \end{cases} \] where \(\theta \in (0, \infty)\). Let \(X_{(2)} = \max(X_1, X_2)\) and \[ \psi(\theta) = P_\theta(X_1 + X_2 < 1), \quad \theta > 0. \] Let \(\delta(X_{(2)})\) be an unbiased estimator of \(\psi(\theta)\) that depends on observations \(X_1\) and \(X_2\) only through \(X_{(2)}\). If \(\delta(t)\) is a continuous function on \((0, \infty)\), then the value of \(18\delta\left(\frac{3}{4}\right)\) equals __________ (answer in integer).

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